15.
Let dλ(
t) be a given nonnegative measure on the real line
, with compact or infinite support, for which all moments
exist and are finite, and μ
0>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes
where σ=σ
n=(
s1,
s2,…,
sn) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness
dmax=2∑
ν=1nsν+2
n−1 if and only if
The proof of the uniqueness of the extremal nodes τ
1,τ
2,…,τ
n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term
R(
f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ
ν, ν=1,2,…,
n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.
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